PHYSICS 2D
PROF. HIRSCH
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QUIZ 5
Formulas:
Time dilation; Length contraction : "t = #"t'$ # "t p ;
L = Lp /#
; c = 3 %10 8 m /s
Lorentz transformation : x'= " (x # vt) ; y' = y ; z' = z ; t'= " (t # vx /c 2 ) ; inverse : v $ -v
uy
ux " v
Velocity transformation : ux '=
; uy '=
; inverse : v $ -v
2
1" ux v /c
# (1" ux v /c 2 )
Spacetime interval : ("s) 2 = (c"t) 2 - ["x 2 + "y 2 + "z 2 ]
Relativistic Doppler shift : f obs = f source 1+ v /c / 1" v /c
r
r
Momentum : p = " mu ; Energy : E = " mc 2 ; Kinetic energy : K = (" #1)mc 2
Rest energy : E 0 = mc 2
;
Electron : me = 0.511 MeV /c 2
E=
p 2c 2 + m 2c 4
Proton : mp = 938.26 MeV /c 2
Neutron : mn = 939.55 MeV /c 2
Atomic mass unit : 1 u = 931.5 MeV /c 2
; electron volt : 1eV = 1.6 "10 -19 J
4
Stefan's law : etot = "T , etot = power/unit area ; " = 5.67 #10$8 W /m 2K 4
#
hc
etot = cU /4 , U = energy density = $ u( ",T)d" ;
Wien's law : "m T =
4.96kB
0
-E/(kB T )
Boltzmann distribution : P(E) = Ce
8$
hc / "
8$f 2
Planck's law : u" ( ",T) = N " ( ") # E ( ",T) = 4 # hc / "kB T
;
N( f ) = 3
"
e
%1
c
Photons : E = hf = pc ; f = c / " ; hc = 12,400 eV A ; k B = (1/11,600)eV /K
Photoelectric effect : eVs = K max = hf " # , # $ work function; Bragg equation : n% = 2d sin &
h
h
Compton scattering : "'- " =
(1 # cos $ ) ;
= 0.0243A
mec
mec
kq q
kq q
kq
Coulomb force : F = 12 2 ; Coulomb energy : U = 1 2 ; Coulomb potential : V =
r
rr
r
r
r r
Force in electric and magnetic fields (Lorentz force) : F = qE + qv " B
Z2
1
Rutherford scattering : "n = C 2
; ke 2 = 14.4 eV A ; hc = 1973 eV A
4
K# sin ($ /2)
1
1
1
1
Hydrogen spectrum :
= R( 2 # 2 )
;
R = 1.097 $10 7 m#1 =
"mn
m
n
911.3A
2
2
2
2
ke Z
Z
ke
me (ke )
mev 2
ke 2 Z
Bohr atom : E n = "
= "E 0 2 ; E 0 =
=
= 13.6eV ; K =
; U ="
2rn
n
2a0
2h 2
2
r
!
hf = E i " E f ; rn = r0 n 2 ; r0 =
!
de Broglie : " =
!
WINTER QUARTER 2012
MARCH 12TH, 2012
h
E
;f =
p
h
a0
Z
; a0 =
h2
= 0.529A ; L = me vr = nh angular momentum
me ke 2
; # = 2$f ; k =
2$
;
"
Wave packets : y(x,t) = $ a j cos(k j x " # j t), or y(x,t) =
E = h# ; p = hk ;
% dk a(k) e
i(kx -# (k )t )
E=
p2
2m
; &k&x ~ 1 ; &#&t ~ 1
j
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d"
"
; vp =
; Heisenberg : #x#p ~ h ; #t#E ~ h
dk
k
E
-i t
h2 " 2#
"#
h
Schrodinger equation : +
U(x)#(x,t)
=
ih
;
#(x,t)
=
$
(x)e
2m "x 2
"t
group and phase velocity : v g =
PHYSICS 2D
PROF. HIRSCH
QUIZ 5
WINTER QUARTER 2012
MARCH 12TH, 2012
%
h 2 # 2$
+
U(x)
$
(x)
=
E
$
(x)
;
dx $ *$ = 1
&
2
2m #x
-%
2 2 2
2
2
n$x
$ hn
h
sin(
) ; En =
;
= 3.81eVA 2 (electron)
2
L
L
2mL
2me
Time " independent Schrodinger equation : -
" square well : # n (x) =
m$ 2
x
2h
1
p2 1
1
; E n = (n + )h$ ; E =
+ m$ 2 x 2 = m$ 2 A 2 ; %n = ±1
2
2m 2
2
!
Harmonic oscillator : "n (x) = H n (x)e
!
Expectation value of[Q] :< Q >= # " * (x)[Q]" (x) dx ; Momentum operator : p =
!
#
Eigenvalues and eigenfunctions : [Q] " = q " (q is a constant) ; uncertainty :
Step potential : reflection coef : R =
!
!
(k1 " k 2 ) 2
, T = 1" R ;
(k1 + k 2 ) 2
k=
h $
i $x
#Q = < Q2 > $ < Q > 2
2m
(E " U)
h2
x2
%
-2 # (x )dx
Tunneling :
" (x) ~ e -# x
;
!
T = e -2#$x ;
T =e
x1
;
# (x) =
2m[U(x) - E]
h2
r
r
r
r -i E t
h2 2
$#
" # + U( r )#( r ,t) = ih
;
#( r ,t) = % ( r )e h
2m
$t
2 2
# h n12 n 22 n 32
3D square well : "(x,y,z) = "1 (x)"2 (y)"3 (z) ; E =
( +
+ )
2m L12 L22 L23
Spherically symmetric potential : "n,l,m l (r,#, $ ) = Rnl (r)Ylm l (#, $ ) ; Ylm l (#, $ ) = Plm l (# )e im l$
r r r
h #
Angular momentum : L = r " p ; [Lz ] =
; [L2 ]Ylm l = l(l +1)h 2Ylm l ; [L z ]Ylm l = ml hYlm l
i #$
ke 2 Z 2
2
2
Radial probability density : P(r) = r | Rnl (r) | ;
Energy : E n = "
2a0 n 2
1 Z
Ground state of hydrogen and hydrogen - like ions : "1,0,0 = 1/ 2 ( ) 3 / 2 e$Zr / a 0
#
a0
! Justify all your answers to all problems. Write clearly.
Schrodinger equation in 3D : -
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Problem 1 (10 points)
The ground state wavefunction for a particle in a harmonic oscillator potential is
2
"(x) = De # $ x with β=4A-2 and D a constant.
(a) Find an expression for D in terms of β.
(b) Find the uncertainty in the position of this particle, Δx, in Angstrom.
(c) Find an expression for the uncertainty in the momentum of this particle, Δp, expressed
in terms of β and h .
(d) If this particle has mass 1973eV/c2, find its average kinetic energy in eV.
$
$
2
&
1 &
" #x 2
=
Use % dx e
and % dx x 2e " #x =
. Do not use any of the formulas for the
#
2# #
"$ !
"$
harmonic oscillator potential, do not use the symbol ω in any of your expressions, and
show all steps in your calculations.
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PHYSICS 2D
PROF. HIRSCH
QUIZ 5
WINTER QUARTER 2012
MARCH 12TH, 2012
Problem 2 (10 points)
10eV
incident
electrons
potential
well
5eV
1A
7eV
transmitted
electrons
2A
Electrons with energy 5eV are incident from the left on a barrier of width 1A and height
10eV. If an electron can go through the barrier it finds a potential well where it stays for a
long time bouncing back and forth (assume its energy doesn't change). It can then escape
from the well through the barrier it came in, or through the barrier on the right side of
height 7eV and width 2A.
(a) What is approximately the probability that an electron incident from the left will be
transmitted through the first barrier and enter the potential well?
(b) For an electron that made it into the potential well, is it more likely to come out
through the right barrier or through the left barrier? Estimate the ratio of the probabilities
(right/left).
(c) Using the results of (a) and (b), estimate how many transmitted electrons will result
for 3000 incident electrons, where "transmitted electrons" are those that went through
both barriers and continue to the right (see figure). From this, deduce the overall
transmission (=# transmitted/# incident) and reflection (# reflected/# incident)
coefficients for this composite barrier. (Note that the sum of both coefficients should be
1).
h2
= 3.81eVA 2 .
Use
2me
!
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Problem 3 (10 points)
The wavefunction for an electron in a hydrogen-like ion with Z=2 is
" (r,# , $ ) = C r e %r / a 0 sin # e %i$
(a) Give the values of the quantum numbers n, l, ml . Justify your answers.
(b) Give the value of the z-component of the angular momentum.
(c) Give the value of the most probable distance of this electron to the nucleus, expressed
in terms of a0. Show your work. Compare your answer with the radius of the orbit in the
Bohr model with the same n and Z. !
Justify all your answers to all problems. Write clearly.